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Ramanujan's sum. In number theory, Ramanujan's sum, usually denoted cq ( n ), is a function of two positive integer variables q and n defined by the formula. where ( a, q) = 1 means that a only takes on values coprime to q . Srinivasa Ramanujan mentioned the sums in a 1918 paper. [ 1]
Ramanujan–Sato series. In mathematics, a Ramanujan–Sato series[ 1][ 2] generalizes Ramanujan ’s pi formulas such as, to the form. by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients , and employing modular forms of higher levels.
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4 . No closed-form expression for the partition function is known, but it has both asymptotic expansions that ...
Chudnovsky algorithm. The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan 's π formulae. Published by the Chudnovsky brothers in 1988, [ 1] it was used to calculate π to a billion decimal places. [ 2]
Srinivasa Ramanujan[ a] (22 December 1887 – 26 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.
1729 (number) 1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is also known as the Ramanujan number or Hardy–Ramanujan number, named after G. H. Hardy and Srinivasa Ramanujan .
See Ramanujan–Sato series. From the mid-20th century onwards, all improvements in calculation of π have been done with the help of calculators or computers. In 1944−45, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were ...
The Ramanujan tau function, studied by Ramanujan ( 1916 ), is the function defined by the following identity: where q = exp (2πiz) with Im z > 0, is the Euler function, η is the Dedekind eta function, and the function Δ (z) is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably ...